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74
1. ALGEBRAIC THEMES
group of rotations of the square card. Show that the bijection
e
$
1
r
$
i
r
2
$ ±
1
r
3
$ ±
i
produces a matching of the multiplication tables of the two groups. That
is, if we apply the bijection to each entry of the multiplication table of
H
, we produce the multiplication table of
R
. Thus, the two groups are
isomorphic.
1.10.4.
Show that the group
C
4
D f
i;
±
1;
±
i;1
g
of fourth roots of unity in
the complex numbers is isomorphic to
Z
4
.
The next several exercises give examples of groups coming from var
ious areas of mathematics and require some topology or real and complex
analysis. Skip the exercises for which you do not have the appropriate
background.
1.10.5.
An
isometry
of
R
3
is a bijective map
T
W
R
3
!
R
3
satisfying
d.T.x/;T.y//
D
d.x;y/
for all
x;y
2
R
3
. Show that the set of isome
tries of
R
3
forms a group. (You can replace
R
3
with any
metric space
.)
1.10.6.
A homeomorphism of
R
3
is a bijective map
T
W
R
3
!
R
3
such
that both
T
and its inverse are continuous. Show that the set of homeomor
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 Fall '08
 EVERAGE
 Algebra, Multiplication

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